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Bow to Your Partner for Some Inverse Square Dancing

When the venerable
Bill Klages
last left the
pages of TV Technology,
he was discussing
some mathematical esoterica
regarding inverse
square law and I was a
bit inspired to pick up,
somewhat, from where
he left off.

We all know inverse square law as the
rule of physics regarding point sources
where the intensity of the source will diminish
by the square of the distance traveled.
For most lighting professionals, this is
a rule to determine fall off of a fixture at a
specific distance.

YOU DO THE MATH
This works two ways. If you have a fixed
relationship between your fixture and your
subject, then using inverse square can aid
in determining the size of fixture necessary
to light your subject to a specific stop.
The opposite of this is if you have a fixture
that is providing too much
intensity, knowing inverse
square can help you quickly
determine how much light
you can reduce by simply increasing
the distance to the
subject to obtain the intensity
you’re looking for.

For the mathematically inclined:

Intensity = 1 divided by the
square of the distance.

Fig. 1: Light rays emanating from a point source diverge out in all directions equally. The further they travel from the source, the more physical area they cover and the more the overall intensity is diminished.

(Click to Enlarge)

Although the actual formula
and hardcore physics only
apply to point sources—that
is sources without lenses to
focus the light, without diffusion,
that derive from a single
point (such as tungsten lamps
or HMI lamps as opposed to fluorescent, a
large light source that is not one point, or
LED, which are multiple point sources), it
can be applied to broader, softer sources as
well—although the math will never be as
clean. Softer sources will have a greater fall
off than just the inverse square.

Looking at some photometrics for
lamps, a typical 1K open-face fixture has
an intensity of 1,000 footcandles of light at
4 feet. If we increase that distance to 8 feet,
our intensity drops to 250 footcandles. We
have doubled our distance, but our intensity
is now 1/4 (or the inverse square the
distance traveled) the output.

If we look at Fig. 1, which is typical to
illustrate inverse square, we see that as
light rays diverge from a point source, they
travel outward in all directions and the further
away from the source the more they
diverge and cover a larger physical area. So
at 1 foot, what might cover a 1-foot square
area, at 2 feet will now cover a 4-foot square
area. At 3 feet, that same light is now spread
out over a 9-foot square area. If we continued
to 4 feet, we’d see the light spreading
over 16 square feet of area. So from 1 foot
to 4 feet, the light has diminished in intensity
1/16 of its initial power because it is
spread over that much more area.

ANIMATED SUBJECTS
When you’re dealing with a fixed subject,
say a news anchor sitting at a desk or
a seated interview subject, the relationship
of that subject to the light source doesn’t
change. If, however, that interview subject
is very animated and leans forward in their chair, their relationship to the light source
can change substantially. This is where understanding
inverse square can help us alleviate
problems before they arise.

If we’re lighting that interviewee with
a 4-foot softbox from four feet away and the
subject leans forward during the interview,
moving one foot closer to the fixture, that
subject has changed their relationship to
that light by 25 percent—or the reverse
of inverse square there’s about a 5 percent
increase in intensity on that subject; definitely a noticeable brightness.

Fig. 2: Illustrating the diminishing intensity of a point source fixture over a given distance. Top numbers represent the theoretical footcandle intensity at that given distance (bottom row of numbers) and the center row of numbers shows the fraction of light – relative to the 1 foot measurement – of light at that point.

(Click to Enlarge)

If, however, you increase the size of
the light source to an 8-foot softbox and
increase the distance to
the subject to 8 feet away,
the subjects movement of
1 foot is now only an increase
in 1 percent—substantially
less.

This is a situation
where bigger is better.
The 4-foot soft source at four feet has the same quality
of light as the 8-foot
source eight feet away, but
the spatial relationship
to the subject is substantially
different.

This also works for
lighting groups of people—
the further away
and larger the source is,
the more evenly the light
will spread among the people.
From one to two feet
away from a fixture there’s
a drop off of 75 percent light intensity, but
from that same source, the distances from
eight to 12 feet (room for three people to
stand) there’s a drop off of less than 1 percent
among them.

If we have the ability to increase that
source to a 20-foot source 20 feet away,
then the intensity difference between our
subjects is absolutely negligible and they
can move within a substantial area without
changing light intensity and exposure.

The practical lesson here is that the
larger and further away the source is, the
larger an area we can cover with relatively
equal illumination from a single source.
This is what I like to call the lighting inverse
square dance!

Jay Holben is the technical editor of
Digital Video and a contributor to Government
Video. He is also the author of
the book, “A Shot in the Dark: A Creative
DIY Guide to Digital Video Lighting on
(Almost) No Budget.”